Discrete para-product operators on variable Hardy spaces

Abstract

Let $p(\cdot):\mathbb R^n\rightarrow(0,\infty)$ be a variable exponent function satisfying the globally log-H\"older continuous condition. In this paper, we obtain the boundedness of para-product operators $\pi_b$ on variable Hardy spaces $H^{p(\cdot)}(\mathbb R^n)$, where $b\in BMO(\mathbb R^n)$. As an application, we show that non-convolution type Calder\'on-Zygmund operators $T$ are bounded on $H^{p(\cdot)}(\mathbb R^n)$ if and only if $T^\ast1=0$, where $\frac{n}{n+\epsilon}<\mbox{essinf}_{x\in\mathbb R^n} p\le \mbox{esssup}_{x\in\mathbb R^n} p\le 1$, $\epsilon$ is the regular exponent of kernel of $T$. Our approach relies on the discrete version of Calder\'on's reproducing formula, discrete Littlewood-Paley-Stein theory and almost orthogonal estimates. These results still hold for variable Hardy space on spaces of homogeneous type by using our methods.